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From Vertices to Fragments
Chapter 7
2Chapter 7 -- From Vertices to Fragments
Part I Objectives
Introduce basic implementation strategies
Clipping Scan conversion
CS 480/680
3Chapter 7 -- From Vertices to Fragments
Overview
At end of the geometric pipeline, vertices have been assembled into primitives
Must clip out primitives that are outside the view frustum Algorithms based on representing
primitives by lists of vertices Must find which pixels can be
affected by each primitive Fragment generation Rasterization or scan conversion
CS 480/680
4Chapter 7 -- From Vertices to Fragments
Required Tasks
Clipping Rasterization or scan conversion Transformations Some tasks deferred until fragment
processing Hidden surface removal Antialiasing
CS 480/680
5Chapter 7 -- From Vertices to Fragments
Rasterization Meta Algorithms
Consider two approaches to rendering a scene with opaque objects
For every pixel, determine which object that projects on the pixel is closest to the viewer and compute the shade of this pixel Ray tracing paradigm
For every object, determine which pixels it covers and shade these pixels Pipeline approach Must keep track of depths
CS 480/680
6Chapter 7 -- From Vertices to Fragments
Clipping 2D against clipping window 3D against clipping volume Easy for line segments polygons Hard for curves and text
Convert to lines and polygons first
CS 480/680
7Chapter 7 -- From Vertices to Fragments
Clipping 2D Line Segments
Brute force approach: compute intersections with all sides of clipping window Inefficient: one division per intersection
CS 480/680
8Chapter 7 -- From Vertices to Fragments
Cohen-Sutherland Algorithm
Idea: eliminate as many cases as possible without computing intersections
Start with four lines that determine the sides of the clipping window
CS 480/680
x = xmaxx = xmin
y = ymax
y = ymin
9Chapter 7 -- From Vertices to Fragments
The Cases
Case 1: both endpoints of line segment inside all four lines Draw (accept) line segment as is
Case 2: both endpoints outside all lines and on same side of a line Discard (reject) the line segment
CS 480/680
x = xmaxx = xmin
y = ymax
y = ymin
10Chapter 7 -- From Vertices to Fragments
The Cases
Case 3: One endpoint inside, one outside Must do at least one intersection
Case 4: Both outside May have part inside Must do at least one intersection
CS 480/680
x = xmaxx = xmin
y = ymax
11Chapter 7 -- From Vertices to Fragments
Defining Outcodes
For each endpoint, define an outcode
Outcodes divide space into 9 regions Computation of outcode requires at
most 4 subtractionsCS 480/680
b0b1b2b3
b0 = 1 if y > ymax, 0 otherwiseb1 = 1 if y < ymin, 0 otherwiseb2 = 1 if x > xmax, 0 otherwiseb3 = 1 if x < xmin, 0 otherwise
12Chapter 7 -- From Vertices to Fragments
Using Outcodes
Consider the 5 cases below AB: outcode(A) = outcode(B) = 0
Accept line segment
CS 480/680
13Chapter 7 -- From Vertices to Fragments
Using Outcodes
CD: outcode (C) = 0, outcode(D) 0 Compute intersection Location of 1 in outcode(D) determines
which edge to intersect with Note if there were a segment from A to
a point in a region with 2 ones in outcode, we might have to do two intersections
CS 480/680
14Chapter 7 -- From Vertices to Fragments
Using Outcodes
EF: outcode(E) logically ANDed with outcode(F) (bitwise) 0 Both outcodes have a 1 bit in the same
place Line segment is outside of
corresponding side of clipping window reject
CS 480/680
15Chapter 7 -- From Vertices to Fragments
Using Outcodes
GH and IJ: same outcodes, neither zero but logical AND yields zero
Shorten line segment by intersecting with one of sides of window
Compute outcode of intersection (new endpoint of shortened line segment)
Reexecute algorithm
CS 480/680
16Chapter 7 -- From Vertices to Fragments
Efficiency
In many applications, the clipping window is small relative to the size of the entire data base Most line segments are outside one or
more side of the window and can be eliminated based on their outcodes
Inefficiency when code has to be re-executed for line segments that must be shortened in more than one step
CS 480/680
17Chapter 7 -- From Vertices to Fragments
Cohen Sutherland in 3D
Use 6-bit outcodes When needed, clip line segment
against planes
CS 480/680
18Chapter 7 -- From Vertices to Fragments
Liang-Barsky Clipping
Consider the parametric form of a line segment
We can distinguish between the cases by looking at the ordering of the values of a where the line determined by the line segment crosses the lines that determine the window
CS 480/680
p(a) = (1-a)p1+ ap2 1 a 0
p1
p2
19Chapter 7 -- From Vertices to Fragments
Liang-Barsky Clipping
In (a): a4 > a3 > a2 > a1
Intersect right, top, left, bottom: shorten
In (b): a4 > a2 > a3 > a1
Intersect right, left, top, bottom: reject
CS 480/680
20Chapter 7 -- From Vertices to Fragments
Advantages
Can accept/reject as easily as with Cohen-Sutherland
Using values of a, we do not have to use algorithm recursively as with C-S
Extends to 3D
CS 480/680
21Chapter 7 -- From Vertices to Fragments
Clipping and Normalization
General clipping in 3D requires intersection of line segments against arbitrary plane
Example: oblique view
CS 480/680
22Chapter 7 -- From Vertices to Fragments
Plane-Line Intersections
CS 480/680
)(
)(
12
1
ppn
ppna o
23Chapter 7 -- From Vertices to Fragments
Normalized Form
Normalization is part of viewing (pre clipping) but after normalization, we clip against sides of right parallelepiped
Typical intersection calculation now requires only a floating point subtraction, e.g. is x > xmax ?
CS 480/680
before normalization after normalization
top view
24Chapter 7 -- From Vertices to Fragments
CS 480/680
25Chapter 7 -- From Vertices to Fragments
Part II Objectives
Introduce clipping algorithms for polygons
Survey hidden-surface algorithms
CS 480/680
26Chapter 7 -- From Vertices to Fragments
Polygon Clipping
Not as simple as line segment clipping Clipping a line segment yields at most
one line segment Clipping a polygon can yield multiple
polygons
However, clipping a convex polygon can yield at most one other polygon
CS 480/680
27Chapter 7 -- From Vertices to Fragments
Tessellation and Convexity
One strategy is to replace nonconvex (concave) polygons with a set of triangular polygons (a tessellation)
Also makes fill easier Tessellation code in GLU library
CS 480/680
28Chapter 7 -- From Vertices to Fragments
Clipping as a Black Box
Can consider line segment clipping as a process that takes in two vertices and produces either no vertices or the vertices of a clipped line segment
CS 480/680
29Chapter 7 -- From Vertices to Fragments
Pipeline Clipping of Line Segments
Clipping against each side of window is independent of other sides Can use four independent clippers in a
pipeline
CS 480/680
30Chapter 7 -- From Vertices to Fragments
Pipeline Clipping of Polygons
Three dimensions: add front and back clippers
Strategy used in SGI Geometry Engine
Small increase in latencyCS 480/680
31Chapter 7 -- From Vertices to Fragments
Bounding Boxes
Rather than doing clipping on a complex polygon, we can use an axis-aligned bounding box or extent Smallest rectangle aligned with axes
that encloses the polygon Simple to compute: max and min of x
and y
CS 480/680
32Chapter 7 -- From Vertices to Fragments
Bounding boxes
Can usually determine accept/reject based only on bounding box
CS 480/680
reject
accept
requires detailed clipping
33Chapter 7 -- From Vertices to Fragments
Clipping and Visibility
Clipping has much in common with hidden-surface removal
In both cases, we are trying to remove objects that are not visible to the camera
Often we can use visibility or occlusion testing early in the process to eliminate as many polygons as possible before going through the entire pipelineCS 480/680
34Chapter 7 -- From Vertices to Fragments
Hidden Surface Removal
Object-space approach: use pair-wise testing between polygons (objects)
Worst case complexity O(n2) for n polygons
CS 480/680
partially obscuring can draw independently
35Chapter 7 -- From Vertices to Fragments
Painter’s Algorithm
Render polygons a back to front order so that polygons behind others are simply painted over
CS 480/680
B behind A as seen by viewer Fill B then A
36Chapter 7 -- From Vertices to Fragments
Depth Sort
Requires ordering of polygons first O(n log n) calculation for ordering Not every polygon is either in front or
behind all other polygons
Order polygons and deal with easy cases first, harder later
CS 480/680
Polygons sorted by distance from COP
37Chapter 7 -- From Vertices to Fragments
Easy Cases
A lies behind all other polygons Can render
Polygons overlap in z but not in either x or y Can render independently
CS 480/680
38Chapter 7 -- From Vertices to Fragments
Hard Cases
CS 480/680
Overlap in all directionsbut one is fully on one side of the other
cyclic overlap
penetration
39Chapter 7 -- From Vertices to Fragments
Back-Face Removal (Culling)
face is visible iff 90 -90 equivalently cos 0 or v • n 0
plane of face has form ax + by +cz +d =0 but after normalization n = ( 0 0 1 0)T
need only test the sign of c In OpenGL we can simply enable culling but
may not work correctly if we have nonconvex objects
CS 480/680
40Chapter 7 -- From Vertices to Fragments
Image Space Approach
Look at each projector (nm for an n x m frame buffer) and find closest of k polygons
Complexity O(nmk) Ray tracing z-buffer
CS 480/680
41Chapter 7 -- From Vertices to Fragments
z-Buffer Algorithm
Use a buffer called the z or depth buffer to store the depth of the closest object at each pixel found so far
As we render each polygon, compare the depth of each pixel to depth in z buffer
If less, place shade of pixel in color buffer and update z buffer
CS 480/680
42Chapter 7 -- From Vertices to Fragments
Efficiency
If we work scan line by scan line as we move across a scan line, the depth changes satisfy ax+by+cz=0
CS 480/680
Along scan line
y = 0z = - x
c
a
In screen space x = 1
43Chapter 7 -- From Vertices to Fragments
Scan-Line Algorithm
Can combine shading and hsr through scan line algorithm
CS 480/680
scan line i: no need for depth information, can only be in noor one polygon
scan line j: need depth information only when inmore than one polygon
44Chapter 7 -- From Vertices to Fragments
Implementation
Need a data structure to store Flag for each polygon (inside/outside) Incremental structure for scan lines
that stores which edges are encountered
Parameters for planes
CS 480/680
45Chapter 7 -- From Vertices to Fragments
Visibility Testing
In many real-time applications, such as games, we want to eliminate as many objects as possible within the application Reduce burden on pipeline Reduce traffic on bus
Partition space with Binary Spatial Partition (BSP) Tree
CS 480/680
46Chapter 7 -- From Vertices to Fragments
Simple Example
CS 480/680
consider 6 parallel polygons
top view
The plane of A separates B and C from D, E and F
47Chapter 7 -- From Vertices to Fragments
BSP Tree
Can continue recursively Plane of C separates B from A Plane of D separates E and F
Can put this information in a BSP tree Use for visibility and occlusion testing
CS 480/680
48Chapter 7 -- From Vertices to Fragments
CS 480/680
49Chapter 7 -- From Vertices to Fragments
Part III Objectives
Survey Line Drawing Algorithms DDA Bresenham
CS 480/680
50Chapter 7 -- From Vertices to Fragments
Rasterization
Rasterization (scan conversion) Determine which pixels that are inside
primitive specified by a set of vertices Produces a set of fragments Fragments have a location (pixel
location) and other attributes such color and texture coordinates that are determined by interpolating values at vertices
Pixel colors determined later using color, texture, and other vertex properties
CS 480/680
51Chapter 7 -- From Vertices to Fragments
Scan Conversion of Line Segments
Start with line segment in window coordinates with integer values for endpoints
Assume implementation has a write_pixel function
CS 480/680
y = mx + h
x
ym
52Chapter 7 -- From Vertices to Fragments
DDA Algorithm
Digital Differential Analyzer DDA was a mechanical device for
numerical solution of differential equations
Line y=mx+ h satisfies differential equation
dy/dx = m = Dy/Dx = y2-y1/x2-x1
Along scan line Dx = 1
CS 480/680
For(x=x1; x<=x2,ix++) { y+=m; write_pixel(x, round(y), line_color)}
53Chapter 7 -- From Vertices to Fragments
Problem
DDA = for each x plot pixel at closest y Problems for steep lines
CS 480/680
54Chapter 7 -- From Vertices to Fragments
Using Symmetry
Use for 1 m 0 For m > 1, swap role of x and y
For each y, plot closest x
CS 480/680
55Chapter 7 -- From Vertices to Fragments
Bresenham’s Algorithm
DDA requires one floating point addition per step
We can eliminate all fp through Bresenham’s algorithm
Consider only 1 m 0 Other cases by symmetry
Assume pixel centers are at half integers If we start at a pixel that has been written,
there are only two candidates for the next pixel to be written into the frame buffer
CS 480/680
56Chapter 7 -- From Vertices to Fragments
Candidate Pixels
CS 480/680
1 m 0
last pixel
candidates
Note that line could havepassed through anypart of this pixel
57Chapter 7 -- From Vertices to Fragments
Decision Variable
CS 480/680
d = Dx(b-a)
d is an integerd > 0 use upper pixeld < 0 use lower pixel
58Chapter 7 -- From Vertices to Fragments
Incremental Form
More efficient if we look at dk, the value of the decision variable at x = k
For each x, we need do only an integer addition and a test
Single instruction on graphics chips
CS 480/680
dk+1= dk –2Dy, if dk <0dk+1= dk –2(Dy- Dx), otherwise
59Chapter 7 -- From Vertices to Fragments
Polygon Scan Conversion
Scan Conversion = Fill How to tell inside from outside
Convex easy Nonsimple difficult Odd even test
Count edge crossings
Winding number
CS 480/680
odd-even fill
60Chapter 7 -- From Vertices to Fragments
Winding Number
Count clockwise encirclements of point
Alternate definition of inside: inside if winding number 0
CS 480/680
winding number = 2
winding number = 1
61Chapter 7 -- From Vertices to Fragments
Filling in the Frame Buffer
Fill at end of pipeline Convex Polygons only Nonconvex polygons assumed to have
been tessellated Shades (colors) have been computed for
vertices (Gouraud shading) Combine with z-buffer algorithm
March across scan lines interpolating shades
Incremental work small
CS 480/680
62Chapter 7 -- From Vertices to Fragments
Using Interpolation
CS 480/680
span
C1
C3
C2
C5
C4
scan line
C1 C2 C3 specified by glColor or by vertex shadingC4 determined by interpolating between C1 and C2
C5 determined by interpolating between C2 and C3
interpolate between C4 and C5 along span
63Chapter 7 -- From Vertices to Fragments
Flood Fill
Fill can be done recursively if we know a seed point located inside (WHITE)
Scan convert edges into buffer in edge/inside color (BLACK)
CS 480/680
flood_fill(int x, int y) { if(read_pixel(x,y)= = WHITE) { write_pixel(x,y,BLACK); flood_fill(x-1, y); flood_fill(x+1, y); flood_fill(x, y+1); flood_fill(x, y-1);} }
64Chapter 7 -- From Vertices to Fragments
Scan Line Fill
Can also fill by maintaining a data structure of all intersections of polygons with scan lines Sort by scan line Fill each span
CS 480/680
vertex order generated by vertex list desired order
65Chapter 7 -- From Vertices to Fragments
Data Structure
CS 480/680
66Chapter 7 -- From Vertices to Fragments
Aliasing
Ideal rasterized line should be 1 pixel wide
Choosing best y for each x (or visa versa) produces aliased raster lines
CS 480/680
67Chapter 7 -- From Vertices to Fragments
Antialiasing by Area Averaging
Color multiple pixels for each x depending on coverage by ideal line
CS 480/680
original antialiased
magnified
68Chapter 7 -- From Vertices to Fragments
Polygon Aliasing
Aliasing problems can be serious for polygons Jaggedness of edges Small polygons neglected Need compositing so colorof one polygon does nottotally determine color ofpixel
CS 480/680
All three polygons should contribute to color
69Chapter 7 -- From Vertices to Fragments
CS 480/680